Equation of Straight Line in Plane/General Equation
Theorem
A straight line $\LL$ is the set of all $\tuple {x, y} \in \R^2$, where:
- $\alpha_1 x + \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.
Proof
Let $y = \map f x$ be the equation of a straight line $\LL$.
From Line in Plane is Straight iff Gradient is Constant, $\LL$ has constant slope.
Thus the derivative of $y$ with respect to $x$ will be of the form:
- $y' = c$
Thus:
\(\ds y\) | \(=\) | \(\ds \int c \rd x\) | Fundamental Theorem of Calculus | |||||||||||
\(\ds \) | \(=\) | \(\ds c x + K\) | Primitive of Constant |
where $K$ is arbitrary.
Taking the equation:
- $\alpha_1 x + \alpha_2 y = \beta$
it can be seen that this can be expressed as:
- $y = -\dfrac {\alpha_1} {\alpha_2} x + \dfrac {\beta} {\alpha_2}$
thus demonstrating that $\alpha_1 x + \alpha_2 y = \beta$ is of the form $y = c x + K$ for some $c, K \in \R$.
$\blacksquare$
Also presented as
Some sources give this as:
- $a x + b y + c = 0$
where $a, b, c \in \R$ are given, and not both $a, b$ are zero.
Its equivalence to the given form can be seen by equating $a = \alpha_1, b = \alpha_2, c = -\beta$.
Also known as
Some sources refer to this equation as a general equation of the first degree.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $3$.
- 1958: P.J. Hilton: Differential Calculus ... (previous) ... (next): Chapter $1$: Introduction to Coordinate Geometry: $(1.2)$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text V$: Vector Spaces: $\S 28$: Linear Transformations
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.7$: General Equation of Line
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): line: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): line: 2.
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $6$: Curves and Coordinates: Cartesian coordinates
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): straight line (in the plane)